Get some proper rim tape, fabric is "best" but can be a little more expensive. Some of the newer flexible plastic ones are quite good. The cheap stiff ones (like a rolled out yoghurt pot) are cack.
Check he is actually pumping them up as hard as he thinks he is. Most track pump pressure gauges are between vague and random. Most accurate i've ever measured was about 5psi out (80 quid track pump tho!). Have seen some that are nearly 30psi out.
At that sort of weight, 20s are far far far too small to grip well/consistently on anything other than bone dry, smooth, clean tarmac. And you'd have to pump them up incredibly hard to use them on anything else, or you'll get snakebites. I used to TT on 18s, when i weighed 9 stone, they were bloody horrible for anything else. (General riding then was on 22 or 23s)
At about 12 stone now, 25s are used for general riding at about 90 psi and 23s at 100 for high days and holidays. (and training camps in the sun, on good roads!)